A new fractional derivative involving the normalized sinc function without singular kernel

0.12 MB·

by  Xiao-Jun Yang

A new fractional derivative involving the
normalized sinc function without singular
kernel
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1 Xiao-Jun Yang1,2, Feng Gao1,2, J. A. Tenreiro Machado3, Dumitru Baleanu4,5
0
2

n
a 1 School of Mechanics and Civil Engineering, China University of Mining and
J
Technology, Xuzhou 221116, China
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2 State Key Laboratory for Geomechanics and Deep Underground Engineering,

]
China University of Mining and Technology, Xuzhou 221116, China
A
C 3 Institute of Engineering, Polytechnic of Porto, Department of Electrical
. Engineering, Rua Dr. Anto´nio Bernardino de Almeida, 4249-015 Porto, Portugal
h
t 4 Department of Mathematics, Cankya University, Ogretmenler Cad. 14,
a
m Balgat-06530, Ankara Turkey
[ 5 Institute of Space Sciences, Magurele-Bucharest, Romania

1
v
0
9
Abstract
5
5
0 In this paper, a new fractional derivative involving the normalized sinc function
.
1 without singular kernel is proposed. The Laplace transform is used to find the ana-
0
lytical solution of the anomalous heat-diffusion problems. The comparative results
7
1 between classical and fractional-order operators are presented. The results are sig-
: nificant in the analysis of one-dimensional anomalous heat-transfer problems.
v
i
X
r Key words: Fractional derivative, anomalous heat diffusion, integral transform,
a
analytical solution
1 Introduction
In recent years, fractional derivatives (FDs) in the sense of Caputo type have
used to describe anomalous behaviors of diffusive phenomena in mathemati-
cal physics involving different kernels, such as the power-law [1], exponential
Corresponding author:E-Mail: [email protected] (X. J. Yang)
∗
Preprint 23 January 2017
[2], Mittag-Leffler [3], stretched exponential [4], and stretched Mittag-Leffler
[5] functions. For example, the fractional diffusion-wave, in the power-law
function kernel was considered in [6]. The numerical solution for the space-
fractional diffusion equation was presented in [7]. The Cauchy problem for the
time-fractional diffusion equations was investigated in [8]. Withthe help ofthe
FD involving the exponential-function kernel, the heat-diffusion problem with
respect to a non-singular fading memory was proposed in [9]. The heat trans-
fer problem within the non-singular second grade fluid was discussed in [10].
The non-singular unsteady flow of the ordinary couple stress fluid was studied
in [11]. With the use of the FD involving the stretched Mittag-Leffler-function
kernel, the Irving–Mullineux oscillator [12] and the Allen-Cahn equation [13]
were also analyzed. For more details see [14,15].
FDs in the sense of Riemann–Liouville type were developed in [16,17,18]. We
can mention the studies about not only the Fokker-Planck [19] and the diffu-
sion [20,21,22,23] equations, but also the wave propagation [24]. Furthermore,
the Chen’s system of the Riemann–Liouville type [25], the monotone itera-
tive method for neutral fractional differential equations [26] and the time-
fractional-order Harry-Dym equation [27] were also discussed. Readers can
find the more details about the distinct versions of FDs in [28].
The normalized sinc function, structured by Whittaker in [29], and its prop-
erties were considered in [30]. Furthermore, the Fourier [31], Laplace [31] and
Sumudu [32] transforms of the NSF were formulated. However, the FD involv-
ing the normalized sinc function without singular kernel has not proposed.
Motivated by the idea, the present article derives a new FD with respect to
the normalized sinc function without singular kernel. Furthermore, based on
the new concept it is considered the applications in one-dimensional anoma-
lous heat-transfer problems.
The structure of the present paper is as follows. In Section 2, a new FD with
respect to the normalized sinc function without singular kernel is presented.
In Section 3, the anomalous heat-diffusion models and their solutions are an-
alyzed by means of the Laplace transform. Finally, the conclusion is outlined
in Section 4.
2 Preliminaries, definitions and integral transforms
In this section, we derive the FD involving the normalized sinc function with-
out singular kernel.
2
2.1 A new FD involving the normalized sinc function without singular kernel
Definition 1 The normalized sinc function is defined by [29,30]:
sin(πx)
sinc(x) = , (1)
πx
where x R.
∈
If ϕ(x) is any smooth function with compact support, where x R, then [30]
∈
1 x sin πx
̟
lim sinc = lim = δ(x), (2)
̟→0 ̟ ̟ ̟→0 π(cid:16)x (cid:17)
(cid:18) (cid:19)
sinc(0) = 1, (3)
where
∞ ϕ(x)sinc x
̟
lim dx = ϕ(0). (4)
̟→0 ̟ (cid:16) (cid:17)
−Z∞
Definition 2 Let Π(x) H1(a,b) and b > a. A new FD involving the
∈
normalized sinc kernel of the function Π(µ) of order ̟ (̟ (0,1)) is defined
∈
as:
µ
̟℘(̟) ̟(µ x)
D(̟)Π(µ) = sinc − Π(1)(x)dx, (5)
a µ 1 ̟ − 1 ̟ !
− Za −
where a ( ,µ), and ℘(̟) is a normalization constant depending on ̟
∈ −∞
such that ℘(0) = ℘(1) = 1.
Following Eq.(1), we obtain
1 µ x sin π(µ−x)
̟
lim sinc − = lim = δ(µ x), (6)
̟→0̟ ̟ ̟→0 π((cid:16)µ x)(cid:17) −
(cid:18) (cid:19) −
where ϕ(x) is any smooth function with compact support where x R such
∈
that
∞
1 x µ
lim ϕ(x) sinc − dx = ϕ(µ). (7)
̟→0 ̟ ̟
−Z∞ (cid:18) (cid:19)
Thus, we have
µ
lim D(̟)Π(µ) = lim ̟℘(̟) sinc ̟(µ−x) Π(1)(x)dx
̟→0a µ ̟→0 (1−̟) a − 1−̟
µ (cid:16) (cid:17)
R
= lim ℘(̟) δ(µ x)Π(1)(x)dx (8)
(cid:18)̟→0 (cid:19)a −
R
= Π(1)(x).
3
When
sin π(µ−x)
̟℘(̟) ̟(µ x) ̟ 1−̟
lim sinc − = lim ℘(̟) (cid:18) ̟ (cid:19) = lim 1,
̟→1 1 ̟ " − 1 ̟ !# ̟→1 1 ̟ π(µ x) ̟→1
− − − −
(9)
we have
µ
lim D(̟)Π(µ) = lim ̟℘(̟) sinc ̟(µ−x) Π(1)(x)dx
̟→1a µ ̟→1 (1−̟) a − 1−̟
µ (cid:16) (cid:17)
R
= lim Π(1)(x)dx (10)
̟→1a
R
= Π(µ) Π(a).
−
For n 1 and ̟ (0,1), the FD D(n+̟)Π(µ) of order (n+ω) is defined as:
≥ ∈ µ
D(n+̟)Π(µ) := D(n) D(̟)Π(µ) . (11)
a µ a µ a µ
(cid:16) (cid:17)
Property 1 (T1) D(̟)θ = 0, where θ is a constant;
0 µ
µ
(T2) D(̟)µ = ̟℘(̟) sinc ̟x dx.
0 µ 1−̟ 1−̟
0
(cid:16) (cid:17)
R
Proof. We have from Eq.(5) that
µ
̟℘(̟) ̟(µ x)
D(̟)θ = sinc − θ(1)dx = 0. (12)
0 µ 1 ̟ − 1 ̟ !
− Z0 −
We have, by using the definition Eq.(5),
µ µ
̟℘(̟) ̟(µ x) ̟℘(̟) ̟x
D(̟)µ = sinc − dx = sinc dx.
0 µ 1−̟ Z0 − 1−̟ ! 1−̟ Z0 (cid:18)1−̟(cid:19)
(13)
2.2 Integral transforms of the new FD involving the normalized sinc function
without singular kernel
Here, we have [31]
sin(πx) 1
sinc(x) = = H(π ξ ) (14)
ℵ{ } ℵ( πx ) s2π −| |
4
such that
sinc ̟x = sin(−1̟−π̟x)
ℵ −1−̟ ℵ −1̟−π̟x
n (cid:16) (cid:17)o (cid:26) (cid:27)
= 1 1−̟H ̟π ξ (15)
− 2π ̟ −1−̟ −| |
= q1 1−̟H (cid:16)̟π + ξ , (cid:17)
2π ̟ 1−̟ | |
q (cid:16) (cid:17)
where is the Fourier transform operator [31], and H (x) is the Heaviside
ℵ
function [31].
The Fourier transform of Eq.(5) can be written as
D(̟)Π(µ)
ℵ 0 µ
n µ o
= ̟℘(̟) sinc ̟(µ−x) Π(1)(x)dx
ℵ( 1−̟ 0 − 1−̟ )
= ̟℘(̟) sRinc (cid:16)̟x (cid:17)Π(1)(x) (16)
1−̟ ℵ −1−̟ ℵ
= ̟℘(̟) n 1 1−(cid:16)̟H ̟(cid:17)πo +nξ [iξΠo(ξ)]
1−̟ 2π ̟ 1−̟ | |
= iξ 1 h℘q(̟)H ̟(cid:16)π + ξ Π(cid:17)(iξ),
2π 1−̟ | |
q (cid:16) (cid:17)
where Π(µ) = Π(ξ).
ℵ{ }
Similarly, we have [31]
sin(πx) 1 π
sinc(x) = = tan−1 (17)
ℑ{ } ℑ( πx ) π (cid:18)s(cid:19)
such that
̟ sin ̟ πx 1 ̟π
sinc x = −1−̟ = tan−1 1−̟ , (18)
ℑ(cid:26) (cid:18)−1−̟ (cid:19)(cid:27) ℑ (cid:16)−1̟−π̟x (cid:17) 1̟−π̟ s !
 
where is the Laplace transform operator [31].
ℑ
From Eq.(18) the Laplace transform of Eq.(5) can be given by:
D(̟)Π(µ)
ℑ 0 µ
=n ̟℘(̟) µsoinc ̟(µ−x) Π(1)(x)dx
ℑ( 1−̟ 0 − 1−̟ ) (19)
R (cid:16) (cid:17)
= ̟℘(̟) sinc ̟x Π(1)(x)
1−̟ ℑ −1−̟ ℑ
̟π
= ℘(̟)tan−n1 1−(cid:16)̟ (sΠ(cid:17)o(s) n Π(0)),o
π s −
(cid:18) (cid:19)
where Π(µ) = Π(s).
ℑ{ }
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As a direct result, we have [32]
sin(πx) tan−1(πζ)
sinc(x) = = (20)
ℜ{ } ℜ( πx ) πζ
such that
̟ sin ̟ πx tan−1 ̟πζ
sinc x = −1−̟ = 1−̟ (21)
ℜ −1 ̟ ℑ (cid:16) ̟πx (cid:17) ̟(cid:16)πζ (cid:17)
(cid:26) (cid:18) − (cid:19)(cid:27)  −1−̟  1−̟ ,
 
where is the Sumudu transform operator [32].
ℜ
Thus, we have from Eq.(13) that
D(̟)Π(µ)
ℜ 0 µ
n µ o
= ̟℘(̟) sinc ̟(µ−x) Π(1)(x)dx
ℜ( 1−̟ 0 − 1−̟ ) (22)
R (cid:16) (cid:17)
= ̟℘(̟) sinc ̟x Π(1)(x)
1−̟ ℜ −1−̟ ℜ
= ℘(̟)tan−n1 ̟π(cid:16)ζ Π(ζ(cid:17))o−Π(0n) , o
πζ 1−̟ ζ
(cid:16) (cid:17)(cid:16) (cid:17)
where Π(µ) = Π(ζ).
ℜ{ }
3 Modelling the anomalous heat-diffusion problems
In this section, we model the anomalous heat-diffusion problems involving
fractional-time and -space derivatives of the normalized sinc function without
singular kernel.
Example 1
The anomalous heat-diffusion within the fractional-time derivative of the nor-
malized sinc function without singular kernel is written as:
∂2Π(µ,τ)
D(̟)Π(µ,τ) = κ , µ > 0, τ > 0, (23)
0 τ ∂µ2
subjected to the initial and boundary conditions:
Π(µ,0) = 0, µ > 0, (24)
Π(0,τ) = λ(τ), τ > 0, (25)
Π(µ,τ) 0, as µ , τ > 0, (26)
→ → ∞
where κ is the thermal diffusivity.
6
With the aid of Eq.(19), Eq.(23) can be transferred into
℘(̟) ̟π d2Π(µ,s)
tan−1 1−̟ (sΠ(µ,s) Π(µ,0)) = κ . (27)
π s ! − dµ2
From Eq.(24) we have the following:
d2Π(µ,s) ℘(̟)s ̟π
= tan−1 1−̟ Π(µ,s), (28)
dµ2 πκ s
!
which leads to
Π(µ,s) = Ω exp µ√H +Ω exp µ√H , (29)
1 2
−
(cid:16) (cid:17) (cid:16) (cid:17)
where Ω and Ω are two unknown constants and
1 2
℘(̟)s ̟π
H = tan−1 1−̟ . (30)
πκ s
!
In view of Eq.(25) and Eq.(26), we have
Ω = 0 (31)
2
such that
Π(µ,s) = λ(s)exp µ√H , (32)
−
where λ(µ) = λ(s). (cid:16) (cid:17)
ℑ{ }
Thus, the Laplace transform solution of Eq.(23) is:
℘(̟)s ̟π
Π(µ,s) = λ(s)exp tan−1 1−̟ µ . (33)
−vu πκ s ! 
u
 t 
Example 2
Theanomalousheat-diffusionwithinthefractional-spacederivative ofthenor-
malized sinc function without singular kernel is
∂Π(µ,τ)
= κ D(1) D(̟)Π(µ,τ) , µ > 0, τ > 0, (34)
∂τ 0 µ 0 µ
(cid:16) (cid:17)
with the initial and boundary conditions:
Π(µ,0) = 0, µ > 0, (35)
Π(0,τ) = λ(τ), τ > 0, (36)
Π(µ,0) 0, as µ , τ > 0, (37)
→ → ∞
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where κ is the thermal diffusivity, and
µ
̟℘(̟) ∂ ̟(µ x)
D(1) D(̟)Π(µ,τ) = sinc − Π(1)(x,τ)dx.
0 µ 0 µ 1 ̟ ∂µ − 1 ̟ !
(cid:16) (cid:17) − Z0 −
(38)
With the help of Eq.(19) and Eq.(35), Eq.(34) can be written as:
s
D(1) D(̟)Π(µ,s) = Π(µ,s), (39)
0 µ 0 µ κ
(cid:16) (cid:17)
where
µ
̟℘(̟) ∂ ̟(µ x)
D(1) D(̟)Π(µ,s) = sinc − Π(1)(x,s)dx. (40)
0 µ 0 µ 1 ̟ ∂µ − 1 ̟ !
(cid:16) (cid:17) − Z0 −
By the integration of Eq.(39) we have
µ µ
̟℘(̟) ̟(µ x) s
D(̟)Π(µ,s) = sinc − Π(1)(x,s)dx = Π(x,s)dx+Θ,
0 µ 1 ̟ − 1 ̟ ! κ
− Z0 − Z0
(41)
where Θ is a constant.
By taking the Sumudu transform operator with µ and Θ = 0, we have
µ µ
̟℘(̟) ̟(µ x) s
sinc − Π(1)(x,s)dx = Π(x,s)dx, (42)
1 ̟ − 1 ̟ ! κ
− Z0 − Z0
which implies that
℘(̟) ̟πζ sζ
tan−1 (Π(ζ,s) Π(0,s)) = Π(ζ,s). (43)
πζ2 1 ̟! − κ
−
From Eq.(35) and Eq.(43), we have the following:
℘(̟) ̟πζ sζ
tan−1 (Π(ζ,s) λ(s)) = Π(ζ,s). (44)
πζ2 1 ̟! − κ
−
Thus, we have
℘(̟)λ(s)tan−1 ̟πζ
Π(ζ,s) = πζ2 1−̟ (45)
℘(̟)tan−1 ̟π(cid:16)ζ (cid:17)sζ
πζ2 1−̟ − κ
where Π(µ,s) = Π(ζ,s) represents the(cid:16)Sum(cid:17)udu transform operator [32].
ℜ{ }
From Eq.(45), the Laplace transform solution of Eq.(23) is:
℘(̟)λ(s)tan−1 ̟πζ
Π(µ,s) = −1 πζ2 1−̟ , (46)
ℜ ℘(̟)tan−1 ̟π(cid:16)ζ (cid:17)sζ
 πζ2 1−̟ − κ 
(cid:16) (cid:17)
 
8
where −1 Π(ζ,s) = Π(µ,s) represents the inverse Sumudu transform op-
ℜ { }
erator [30].
When ̟ = 0, Eq.(33) and Eq.(46) become the Laplace transform solution of
the classical heat-diffusion equation [33]:
s
Π(µ,s) = λ(s)exp µ . (47)
− κ
(cid:18) r (cid:19)
which is in agreement with the result in [31].
4 Conclusions
In the present study, we addressed a new FD in respect to the normalized sinc
function without singular kernel. Moreover, the Fourier, Laplace and Sumudu
transforms of the FD operator and the Laplace–transform solutions of the
anomalous heat-diffusion equations were considered. The analytical solutions
of the classical and anomalous heat-diffusion equations in the form of the
Laplace transform were also compared. The new formulation may be used to
support a new perspective for describing the anomalous behaviors in mathe-
matical physics.
References
[1] Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency
independent–II. Geophysical Journal International, 13(5), 529-539.
[2] Caputo, M., Fabrizio, M. A., A new definition of fractional derivative without
singular kernel, Progress in Fractional Differentiation and Applications, 1
(2015), 2, pp.73–85.
[3] Yang,X.J.(2016).Fractionalderivativesofconstantandvariableordersapplied
to anomalous relaxation models in heat-transfer problems, Thermal Science,
DOI: 10.2298/TSCI161216326Y.
[4] Sun,H., Hao, X., Zhang,Y., Baleanu, D., Relaxation anddiffusion models with
non-singular kernels, Physica A, 468, 2017, 590-596.
[5] Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and
non-singular kernel: Theory and application to heat transfer model, Thermal
Science, 20 (2) (2016), 763-769.
[6] Mainardi, F. (1996). Fractional relaxation-oscillation and fractional diffusion-
wave phenomena. Chaos, Solitons & Fractals, 7(9), 1461-1477.
9
[7] Tadjeran, C., Meerschaert, M. M., Scheffler, H. P. (2006). A second-order
accurate numerical approximation for the fractional diffusion equation. Journal
of Computational Physics, 213(1), 205-213.
[8] Scalas, E., Gorenflo, R., Mainardi, F., & Raberto, M. (2003). Revisiting the
derivation of the fractional diffusion equation. Fractals, 11, 281-289.
[9] Hristov, J. (2016). Transient heat diffusion with a non-singular fading
memory, Thermal Science, 20(2), 757-762.
[10] Shah, N. A., & Khan, I. (2016). Heat transfer analysis in a second grade fluid
over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives.
The European Physical Journal C, 76(7), 1-11.
[11] Akhtar, S. (2016). Flows between two parallel plates of couple stress fluids
with time-fractional Caputo and Caputo-Fabrizio derivatives. The European
Physical Journal Plus, 131(11), 401.
[12] Go´mez-Aguilar, J. F. (2017). Irving–Mullineux oscillator via fractional
derivatives with Mittag-Leffler kernel. Chaos, Solitons & Fractals, 95, 179-186.
[13] Algahtani, O. J. J. (2016). Comparing the Atangana–Baleanu and Caputo–
Fabrizio derivative with fractional order: Allen Cahn model. Chaos, Solitons &
Fractals, 89, 552-559.
[14] Kochubei, A. N. (2011). General fractional calculus, evolution equations, and
renewal processes. Integral Equations and Operator Theory, 71(4), 583-600.
[15] Luchko,Y.,&Yamamoto,M.(2016).Generaltime-fractionaldiffusionequation:
some uniqueness and existence results for the initial-boundary-value problems.
Fractional Calculus and Applied Analysis, 19(3), 676-695.
[16] West, B.J.,Fractional CalculusView ofComplexity: Tomorrow’s Science,CRC
Press, Boca Raton, 2015.
[17] Sabatier, J., Agrawal, O. P., Machado, J. T. (2007). Advances in fractional
calculus, Springer.
[18] Yang, X. J., A new fractional derivative without singular kernel: Application to
the modellingof thesteady heat flow,ThermalScience, 20(2016), 2,pp.753-756
[19] Metzler, R., Barkai, E., & Klafter, J. (1999). Anomalous diffusion and
relaxation close to thermal equilibrium: a fractional Fokker-Planck equation
approach. Physical Review Letters, 82(12), 3563.
[20] Mainardi, F., Pagnini, G., & Gorenflo, R. (2007). Some aspects of fractional
diffusion equations of single and distributed order. Applied Mathematics and
Computation, 187(1), 295-305.
[21] Chen, C. M., Liu, F., Turner, I., & Anh, V. (2007). A Fourier method for the
fractional diffusionequation describingsub-diffusion.Journalof Computational
Physics, 227(2), 886-897.
10